Abstract: For any given field $F$ there is a well known parametrizing space for elementary p-abelian Galois extensions of $F$; for example, if $K$ contains a primitive pth root of unity, Kummer theory provides this parametrizing space for us. By putting additional structure on these parametrizing spaces, we are able to give a parametrizing space for solutions to any given embedding problem where the quotient is a cyclic $p$-group and the kernel is an elementary $p$-abelian group. This allows us to give an explicit count to the number of such solutions, and in particular we can make certain universal statements about the number of solutions to such embedding problems. For instance, we use our results to show that $p$-groups have unbounded realization multiplicity.